1. Chapter 32 Production We now add production into the economy. Two goods: foods and clothes (F and C). Two inputs: L and K. Two firms: one producing F and the other producing C. Consumers are endowed with factor inputs and ownership of firms (shareholders). So the profit of firms will be distributed to consumers.

2. First look at the Pareto optimum. Efficiency in the use of inputs in production requires that in an Edgeworth box (where the endowed labor and the endowed capital are the width and length of the box respectively), two firms’ MRTS must equal. Suppose MRTS L,K F =2>MRTS L,K C =0.5, then one unit of L should be moved from C to F and one unit of K from F to C. Both C and F can be increased.

3. We can plot all the points in the Edgeworth box into the plane of F-C and we can get the production possibility set which shows the various combinations of foods and clothes that can be produced with fixed inputs. The production possibility frontier then corresponds to the Pareto set. If we increase one unit of F, how many units of C do we have to decrease?

5. This is called the marginal rate of transformation. By definition, it is ∆C/∆F. By moving one unit of L from C to F, we increase F by MPL F and decrease C by MPL C. Hence, MRT F,C = MPL C / MPL F. Alternatively, we can move one unit of K from C to F, then we increase F by MPK F and decrease C by MPK C. Hence, MRT F,C = MPK C / MPK F. However, since at Pareto set, MPL C / MPK C = MPL F / MPK F, MPL C / MPL F = MPK C / MPK F, so both gives you the same MRT.

6. Efficiency in exchange still has to hold. In other words, given F and C are produced, efficiency implies that MRS A F,C =MRS B F,C. Efficiency in the output market means producers produce what consumers want. This implies MRT F,C =MRS A F,C =MRS B F,C. Suppose MRT F,C =0.5<MRS A F,C =2, then we can decrease C by 0.5 and increase F by 1.

7. This change makes consumer A strictly better off since increasing F by 1, he is willing to give up C by 2 and now the society only gives up C by 0.5. How does market achieve this? Efficiency in the use of inputs in production is achieved because firms max profits implies they min costs. Therefore, their isoquants are tangent to isocost. Since the isocosts of firms have the same slope. We get:

9. MRTS L,K F =w/r=MRTS L,K C. Efficiency in exchange is achieved by the same reason as in the last chapter, i.e., MRS A F,C =p F /p C =MRS B F,C. Efficiency in the output market is achieved because when each firm (say firm F) max profit, it chooses p F MPL F =w and p C MPL C =w. Therefore, MRT F,C = MPL C /MPL F = p F /p C. Since MRS A F,C =p F /p C =MRS B F,C, so MRT=MRS.

10. Cheese (1 lb) Wine (1 gal) Holland1 (1/2W*)2 (2C) Italy6 (2W)3 (1/2 C*) hours of labor required to produce

11. In each box, we have the hours of labor required to produce a certain amount of output. We can draw the production possibility frontier of each country. Now what is the production frontier of the world? To produce the first unit of C, should ask Holland to produce. Up to the point where all the labor of Holland has been employed to produce C. Mention international trade.

12. Mention the Walras’ law.  F = p F F-wL F -rK F and  C = p C C-wL C -rK C. p F x F A +p C x C A =wL A +rK A + θ F  F + θ C  C and. p F x F B +p C x C B =wL B +rK B +(1-θ F )  F +(1- θ C )  C. Adding these up, we get p F (x F A +x F B - F)+p C (x C A +x C B -C)+w(L F +L C -L A - L B )+r(K F +K C -K A -K B )+  F -(θ F  F +(1- θ F )  F )+  C -(θ C  C +(1-θ C )  C )=0.

13. To summarize: Efficiency in exchange: MRS A F,C =p F /p C =MRS B F,C. Efficiency in the use of inputs in production: MRTS L,K F =w/r=MRTS L,K C. Efficiency in the output market: MRT F,C = MPL C /MPL F = p F /p C = MRS A F,C.